Are these true for a martingale? $Eleft[ frac{X_{n+1}}{X_n} right] = 1, Eleft[ frac{X_{n+2}}{X_n} right] = 1 $

Question

Let $(X_n)_{n in mathbb{Z}_+}$
be a martingale, $X_n(omega) neq 0$, and $X_{n+1}/X_n, X_{n+2}/X_n in L^1 (n in mathbb{Z}_+)$
Do the following hold for $n in mathbb{Z}_+$?
$Eleft[ frac{X_{n+1}}{X_n} right] = 1, Eleft[ frac{X_{n+2}}{X_n} right] = 1 $

Kavi Rama Murthy · Answer

$E(X_{n+m}|mathcal F_m)=X_m$. This implies $E(frac {X_{n+m}} {X_m}|mathcal F_m)=1$ provided $frac {X_{n+m}} {X_m}$ is integrable. Taking expectation on both sides we get $E(frac {X_{n+m}} {X_m})=1$ .

Answered by Kavi Rama Murthy on December 23, 2021

Are these true for a martingale? $Eleft[ frac{X_{n+1}}{X_n} right] = 1, Eleft[ frac{X_{n+2}}{X_n} right] = 1 $

One Answer

Add your own answers!

Ask a Question